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Simple Linear Regression: Model, Equation, and Least Squares Method

Learn about simple linear regression, including the regression model, equation, and the least squares method. Understand how to estimate the regression line and make predictions.

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Simple Linear Regression: Model, Equation, and Least Squares Method

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  1. Slides by JOHN LOUCKS St. Edward’s University

  2. Chapter 14, Part ASimple Linear Regression • Simple Linear Regression Model • Least Squares Method • Coefficient of Determination

  3. Simple Linear Regression • Managerial decisions often are based on the relationship between two or more variables. • Regression analysis can be used to develop an equation showing how the variables are related. • The variable being predicted is called the dependent variable and is denoted by y. • The variables being used to predict the value of the dependent variable are called the independent variables and are denoted by x.

  4. Simple Linear Regression • Simple linear regression involves one independent variable and one dependent variable. • The relationship between the two variables is approximated by a straight line. • Regression analysis involving two or more independent variables is called multiple regression.

  5. Simple Linear Regression Model • The equation that describes how y is related to x and • an error term is called the regression model. • The simple linear regression model is: y = b0 + b1x +e where: • b0 and b1 are called parameters of the model, • e is a random variable called the error term.

  6. Simple Linear Regression Equation • The simple linear regression equation is: E(y) = 0 + 1x • Graph of the regression equation is a straight line. • b0 is the y intercept of the regression line. • b1 is the slope of the regression line. • E(y) is the expected value of y for a given x value.

  7. E(y) x Simple Linear Regression Equation • Positive Linear Relationship Regression line Intercept b0 Slope b1 is positive

  8. E(y) x Simple Linear Regression Equation • Negative Linear Relationship Intercept b0 Regression line Slope b1 is negative

  9. E(y) x Simple Linear Regression Equation • No Relationship Intercept b0 Regression line Slope b1 is 0

  10. is the estimated value of y for a given x value. Estimated Simple Linear Regression Equation • The estimated simple linear regression equation • The graph is called the estimated regression line. • b0 is the y intercept of the line. • b1 is the slope of the line.

  11. Sample Data: x y x1 y1 . . . . xnyn Estimated Regression Equation Sample Statistics b0, b1 Estimation Process Regression Model y = b0 + b1x +e Regression Equation E(y) = b0 + b1x Unknown Parameters b0, b1 b0 and b1 provide estimates of b0 and b1

  12. ^ yi = estimated value of the dependent variable for the ith observation Least Squares Method • Least Squares Criterion where: yi = observed value of the dependent variable for the ith observation

  13. _ _ x = mean value for independent variable y = mean value for dependent variable Least Squares Method • Slope for the Estimated Regression Equation where: xi = value of independent variable for ith observation yi = value of dependent variable for ith observation

  14. Least Squares Method • y-Intercept for the Estimated Regression Equation

  15. Simple Linear Regression • Example: Reed Auto Sales Reed Auto periodically has a special week-long sale. As part of the advertising campaign Reed runs one or more television commercials during the weekend preceding the sale. Data from a sample of 5 previous sales are shown on the next slide.

  16. Simple Linear Regression • Example: Reed Auto Sales Number of TV Ads (x) Number of Cars Sold (y) 1 3 2 1 3 14 24 18 17 27 Sx = 10 Sy = 100

  17. Estimated Regression Equation • Slope for the Estimated Regression Equation • y-Intercept for the Estimated Regression Equation • Estimated Regression Equation

  18. Using Excel’s Chart Tools for Scatter Diagram & Estimated Regression Equation • Excel Worksheet (showing data)

  19. continue Using Excel’s Chart Tools for Scatter Diagram & Estimated Regression Equation • Producing a Scatter Diagram Step 1 Select cells B1:C6 Step 2 Click the Insert tab on the Excel ribbon Step 3 In the Charts group, click Scatter Step 4 When the list of scatter diagram subtypes appears: Click Scatter with only Markers Step 5 In the Chart Layouts group, click Layout 1 Step 6 Select the Chart Title and replace it with Reed Auto Sales Estimated Regression Equation

  20. Using Excel’s Chart Tools for Scatter Diagram & Estimated Regression Equation • Producing a Scatter Diagram Step 7 Select the Horizontal Axis Title and replace it with TV Ads Step 8 Select the Vertical Axis Title and replace it with Cars Sold Step 9 Right click on the legend and click Delete Step 10 Position the mouse pointer over any Vertical Axis Major Gridline in the scatter diagram and right-click to display a list of options and then choose Delete

  21. Using Excel’s Chart Tools for Scatter Diagram & Estimated Regression Equation • Adding the Trendline Step 11 Position the mouse pointer over any data point in the scatter diagram and right-click to display a list of options Step 12 Choose Add Trendline Step 13 When the Format Trendline dialog box appears: Select Trendline Options and then Choose Linear from the Trend/Regression Type list Choose Display Equation on chart Click Close

  22. Reed Auto Sales Estimated Regression Line Using Excel’s Chart Tools for Scatter Diagram & Estimated Regression Equation

  23. Coefficient of Determination • Relationship Among SST, SSR, SSE SST = SSR + SSE where: SST = total sum of squares SSR = sum of squares due to regression SSE = sum of squares due to error

  24. Coefficient of Determination • The coefficient of determination is: r2 = SSR/SST where: SSR = sum of squares due to regression SST = total sum of squares

  25. Coefficient of Determination r2 = SSR/SST = 100/114 = .8772 The regression relationship is very strong; 87.72% of the variability in the number of cars sold can be explained by the linear relationship between the number of TV ads and the number of cars sold.

  26. Using Excel to Compute the Coefficient of Determination • Displaying the Coefficient of Determination Step 1 Position the mouse pointer over any data point in the scatter diagram and right-click to display a list of options Step 2 Choose Add Trendline Step 3 When the Trendline dialog box appears: Select Trendline Options and then Choose Display R-squared value on chart Click Close

  27. Reed Auto Sales Estimated Regression Line Using Excel to Compute the Coefficient of Determination

  28. where: b1 = the slope of the estimated regression equation Sample Correlation Coefficient

  29. The sign of b1 in the equation is “+”. Sample Correlation Coefficient rxy = +.9366

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